Many physical systems can be modelled as parameter-dependent variational problems. In numerous cases, multiple equilibria co-exist, requiring the evaluation of their stability, and the monitoring of transitions between them. Generally, the stability characteristics of the equilibria change near folds in the parameter space. The direction of stability changes is embedded in a specific projection of the solutions, known as distinguished bifurcation diagrams. In this article, we identify such projections for variational problems characterized by fixed-free ends - a class of problems frequently encountered in mechanics. Using these diagrams, we study an Elastica subject to an end load applied through a rigid lever arm. Several instances of snap-back instability are reported, along with their dependence on system parameters through numerical examples. These findings have potential applications in the design of soft robot arms and other actuator designs.

翻译：许多物理系统可建模为参数依赖的变分问题。在众多情况下，多个平衡态共存，需要评估其稳定性并监测它们之间的转变。通常，平衡态的稳定性特征在参数空间的折叠点附近发生变化。稳定性变化的方向蕴含于解的特定投影中，即所谓的显著分岔图。本文针对具有固定-自由端特征的变分问题——这类问题在力学中频繁出现——识别了此类投影。利用这些分岔图，我们研究了通过刚性杠杆臂施加末端载荷的弹性杆系统。通过数值算例，报告了若干回弹失稳现象及其对系统参数的依赖性。这些发现对软体机械臂及其他执行器设计具有潜在应用价值。